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Properties

Label 14.14.7420033802...1232.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{21}\cdot 29^{12}$
Root discriminant $50.70$
Ramified primes $2, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-17, 338, 2429, 2284, -6342, -4278, 5734, 2584, -2333, -640, 450, 62, -37, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 37*x^12 + 62*x^11 + 450*x^10 - 640*x^9 - 2333*x^8 + 2584*x^7 + 5734*x^6 - 4278*x^5 - 6342*x^4 + 2284*x^3 + 2429*x^2 + 338*x - 17)
 
gp: K = bnfinit(x^14 - 2*x^13 - 37*x^12 + 62*x^11 + 450*x^10 - 640*x^9 - 2333*x^8 + 2584*x^7 + 5734*x^6 - 4278*x^5 - 6342*x^4 + 2284*x^3 + 2429*x^2 + 338*x - 17, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 37 x^{12} + 62 x^{11} + 450 x^{10} - 640 x^{9} - 2333 x^{8} + 2584 x^{7} + 5734 x^{6} - 4278 x^{5} - 6342 x^{4} + 2284 x^{3} + 2429 x^{2} + 338 x - 17 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(742003380228915810271232=2^{21}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.70$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(232=2^{3}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{232}(1,·)$, $\chi_{232}(197,·)$, $\chi_{232}(165,·)$, $\chi_{232}(65,·)$, $\chi_{232}(161,·)$, $\chi_{232}(141,·)$, $\chi_{232}(45,·)$, $\chi_{232}(81,·)$, $\chi_{232}(181,·)$, $\chi_{232}(169,·)$, $\chi_{232}(25,·)$, $\chi_{232}(49,·)$, $\chi_{232}(53,·)$, $\chi_{232}(117,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17} a^{10} - \frac{5}{17} a^{9} - \frac{7}{17} a^{8} + \frac{1}{17} a^{7} - \frac{7}{17} a^{4} + \frac{1}{17} a^{3} - \frac{1}{17} a^{2} + \frac{5}{17} a$, $\frac{1}{17} a^{11} + \frac{2}{17} a^{9} + \frac{5}{17} a^{7} - \frac{7}{17} a^{5} + \frac{4}{17} a^{3} + \frac{8}{17} a$, $\frac{1}{17} a^{12} - \frac{7}{17} a^{9} + \frac{2}{17} a^{8} - \frac{2}{17} a^{7} - \frac{7}{17} a^{6} + \frac{1}{17} a^{4} - \frac{2}{17} a^{3} - \frac{7}{17} a^{2} + \frac{7}{17} a$, $\frac{1}{12794311985681} a^{13} + \frac{208169850339}{12794311985681} a^{12} + \frac{18799820314}{752606587393} a^{11} - \frac{232361762071}{12794311985681} a^{10} - \frac{4452937222944}{12794311985681} a^{9} + \frac{1164766358101}{12794311985681} a^{8} + \frac{4301597934804}{12794311985681} a^{7} + \frac{4765674360786}{12794311985681} a^{6} + \frac{137907297644}{12794311985681} a^{5} + \frac{5477277152127}{12794311985681} a^{4} + \frac{4248180905098}{12794311985681} a^{3} - \frac{381451018248}{12794311985681} a^{2} - \frac{5686566882170}{12794311985681} a - \frac{273459474355}{752606587393}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 36310148.8597 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{2}) \), 7.7.594823321.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.14.21.34$x^{14} + 4 x^{13} + 8 x^{12} + 4 x^{11} + 5 x^{10} + 8 x^{9} - 6 x^{8} - 6 x^{7} + x^{6} + 6 x^{5} + 2 x^{3} + 7 x^{2} + 6 x - 7$$2$$7$$21$$C_{14}$$[3]^{7}$
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$